Hi, and thank you for your reply! Very appreciated!
There are in fact only two choices when choosing alternative ways to log in.
One is to continue to use the "preferred method" which is Microsoft Authenticator, and the other is "Switch to Microsoft Authenticator"... go figure...
Well, the solution for me was for the domain administrator to disable 2FA on my account, and re-enable it, to reset it. Then I could register the app correctly again on my new phone, and all is well and wonderful in Microsoft-land again :)
I have just enabled backup to cloud now, so hopefully I will not hit the same wall again in the future :)
Tormod